\newproblem{lay:4_4_8}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.4.8}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Find the coordinates of $\mathbf{x}=(0,0,-2)$ relative to the basis $B=\{(1,1,3),(2,0,8),(1,-1,3)\}$
}{
  % Solution
	The coordinates of $\mathbf{x}$ in the basis $B$ specify the linear combination of the vectors in the basis $B$ to find $\mathbf{x}$. We need to find
	the weights such that 
	\begin{center}
		$\mathbf{x}=x_1\mathbf{b}_1+x_2\mathbf{b}_2+x_3\mathbf{b}_3=x_1\begin{pmatrix}1\\1\\3\end{pmatrix}+x_2\begin{pmatrix}2\\0\\8\end{pmatrix}
		   +x_3\begin{pmatrix}1\\-1\\3\end{pmatrix}$
	\end{center}
	We can solve this problem through the augmented matrix
	\begin{center}
		$\left(\begin{array}{rrr|r} 1 & 2 & 1 & 0\\ 1 & 0 & -1 & 0\\ 3 & 8 & 3 &-2\end{array}\right) \sim
		 \left(\begin{array}{rrr|r} 1 & 0 & 0 & 1\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 1\end{array}\right)$
	\end{center}
	The coordinates are $[\mathbf{x}]_B=(1,-1,1)$.
}
\useproblem{lay:4_4_8}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
